Its fair to say that tri-bars revolutionised cycling. During the
1980s, despite the advent of low profile bicycles, disk wheels, skin suits,
aero helmets (and ever increasing traffic flow) virtually no time trialling
records were broken. Even the famed World Hour Record remained undisturbed.
Since tri bars came along in the early 1990s records have fallen regularly
to riders with all types of frame (from round steel tubes to carbon monocoques) and all types
of wheel (disks and spoked). The fundamental difference has been the direct
support of the body either on tri bars or, even more directly, in the Graeme
Obree position.
Tri bars are also known as aero bars though many tri-bar positions are far from aerodynamic. At best they do allow the rider to get a low position with horizontal forearms, though this much is achievable with ordinary dropped handlebars. The narrow hand position improves aerodynamics.
For most users this is the most important advantage though they dont always realise it. Relaxed muscles disperse the lactate which builds up in working ones. Tri bars, by keeping he shoulders and arms relaxed, allow the legs to work at a rate which would otherwise not always be possible - especially in a forward position.
The orthodox cycling position described previously (knee over pedal) restricts power to all but the most technically gifted. The amount of power that a rider can put into the pedals is, ultimately, limited by body weight. Tri bars allow you to bring more of your bodyweight into play by supporting your body in a forward position.
With the bodyweight forward there are most of the disadvantages of any steep position - compromised stability, control, handling and traction. The combination of tri-bars and an orthodox position has very few disadvantages - response time for braking, perhaps, and the nuisance factor and weight of the tri-bars when not using them.
A slightly forward position (up to about 80 degrees seat angle) offers improved power and takes advantage of tri-bars. What about moving forward even further? The problem with moving forward further is that the weight moves entirely over the handlebars and it becomes impossible to keep it over the pedals - pressing down hard results in the bottom lifting off the saddle. There are two ways that riders tend to deal with this - they either raise the saddle to reach their bottoms or raise the handle bars (Spencer Smith style) to get a more upright position.